Download Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation Dorit Aharonov

Adiabatic Quantum Computation is Equivalent to Standard
Quantum Computation
Dorit Aharonov∗
Wim van Dam†
Julia Kempe‡
Zeph Landau§
Seth Lloyd¶
arXiv:quant-ph/0405098 v1 18 May 2004
Oded Regevk
July 10, 2004
Abstract
Adiabatic quantum computation has recently attracted attention in the physics and computer science
communities, but its computational power has been unknown. We settle this question and describe an
efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum circuit model are polynomially equivalent. Our result
can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with
nearest neighbor interactions. The equivalence between the models provides a new vantage point from
which to tackle the central issues in quantum computation, namely designing new quantum algorithms
and constructing fault tolerant quantum computers. In particular, by translating the main open questions
in quantum algorithms to the language of spectral gaps of sparse matrices, the result makes quantum
algorithmic questions accessible to a wider scientific audience, acquainted with mathematical physics,
expander theory and rapidly mixing Markov chains.
∗
Computer Science Division, UC Berkeley, CA 94720, and Department of Computer Science and Engineering, Hebrew University, Jerusalem, Israel 91904. [email protected].
†
Department of Physics, MIT, Cambridge, MA 02139. [email protected]
‡
Computer Science Division and Department of Chemistry, UC Berkeley, CA 94720, and CNRS-LRI UMR 8623, Université
de Paris-Sud, Orsay, France 91405. [email protected]
§
Department of Mathematics, City College of New York. [email protected]
¶
Department of Mechanical Engineering, MIT, Cambridge, MA 02139. [email protected]
k
EECS Department, UC Berkeley, Berkeley, CA 94720. [email protected]
1 Introduction
Quantum computation has emerged in the last decade as an exciting and promising direction of research
due to several breakthrough discoveries. Shor’s quantum algorithm for factorization [1], followed by several other algorithms to solve algebraic and combinatorial problems (see, e.g., [2–5]) have demonstrated the
possible exponential advantage of quantum computing systems over classical ones. These discoveries motivated interest in the physical implementation of quantum computation, resulting (to date) in the realization
of small-scale quantum computations in various systems (see, e.g., [6–12]). The field now faces two major
challenges. The first is to extend the capabilities of quantum algorithms, and in particular, to come up with
substantially new algorithmic techniques which go beyond the main quantum tool: the Fourier transform.
The second is to improve the current experimental and theoretical methods [13] of protecting the computation against decoherence and errors, thereby addressing the main stumbling blocks in the realization of
large-scale quantum computers. Recently, Farhi et al. [14,15] suggested an ingenious paradigm for quantum
algorithms called adiabatic computation, which attracted considerable attention [14–21] since it was shown
to exhibit promising algorithmic capabilities [16–18], as well as inherent robustness against decoherence
and control errors [20].
In the adiabatic paradigm, a combinatorial problem is rephrased as the problem of finding the lowest
energy state (namely, the ground state) of a “target” Hamiltonian Hfinal (a Hamiltonian is simply a Hermitian matrix). For the Hamiltonian to be physically realistic, we require that it is local, i.e., involves only
interactions between a constant number of particles. To solve the combinatorial problem, a quantum system
is initialized in the ground state of an initial local Hamiltonian Hinit , and then Hinit is slowly transformed
into Hfinal . A celebrated theorem from physics called the quantum adiabatic theorem [22, 23] implies that if
the transformation is carried out sufficiently slowly, the system tracks the ground state of the time varying
Hamiltonian and therefore ends up in the desired ground state of Hfinal . Indeed, if the spectral gap of the
time varying Hamiltonian is never too small then the entire process can be carried out efficiently. Adiabatic computation can therefore be viewed as a process that takes a quantum state to another. We remark
that previous research in the area followed [15] and focused on the case where Hfinal encodes a classical
optimization problem and so is diagonal. This means that the final state is a basis state. We observe here
that the relevant and natural model from a physical point of view does not require this. Using the same
physical resources (namely, local Hamiltonians) one can actually adiabatically generate much more complicated superpositions. Adiabatic computation is thus viewed as a process that takes a tensor product state to
a (possibly more general) quantum superposition.
The focus of this paper is the characterization of the computational power of such adiabatic computations. [21] began to address this question when they showed that a related model to adiabatic computation,
which used simulatable Hamiltonians (i.e., Hamiltonians which are given non explicitly as the result of
applying sequences of quantum gates) is as powerful as conventional quantum circuits. [21] left the main
question of universality of adiabatic computation with physically realistic (namely, local) Hamiltonians,
open. This model might seem at first sight less powerful than standard quantum computation, because of
the local, almost explicit, specification of the final state. Indeed, previous results gave only partial answers
regarding the computational capabilities of this model. [15] used adiabatic computation to tackle an NPcomplete problem. There is mounting evidence [17, 19, 24] that the algorithm of [15] takes exponential time
in the worst case for such problems. [19, 25] showed that adiabatic computation can implement Grover’s
quantum search algorithm [26], and [16–18] showed that adiabatic computation can ’tunnel’ through wide
energy barriers and thus outperform simulated annealing, the classical counterpart of the adiabatic model.
These results demonstrated polynomial speed-ups of adiabatic computation over general classical search algorithms 1 . From the other direction, [15] imply that adiabatic computation is not stronger computationally
1
In fact, [16, 17] even demonstrated an exponential speed-up of adiabatic computation over classical local search algorithms
1
than the standard quantum model. Where exactly does the adiabatic model sit on the scale between polynomial advantage over classical computation, and the full quantum computational power, was unclear. In fact,
even the question of whether adiabatic computers can simulate general classical computations efficiently
was unknown.
1.1 Results
Our main result clarifies the picture. We show:
Theorem 1. The model of adiabatic computation with local Hamiltonians involving three qubit interactions
is polynomially equivalent to the standard model of quantum computation.
This shows that universal quantum computation can be fully studied and implemented in the adiabatic
framework, and so adiabatic computation can be thought of as an alternative model to quantum computation.
We also show that a similar theorem holds when we relax the requirement that the Hamiltonians be
local, and allow sparse Hamiltonians. These are Hermitian matrices which have at most polynomially many
non zero elements at each row and column, where we also require that the matrix is explicit, namely each
element can be computed efficiently given its indexes. We show
Corollary 2. The model of adiabatic computation with sparse Hamiltonians is polynomially equivalent to
the standard model of quantum computation,
Corollary 2 is potentially more useful in the context of algorithms and complexity, since sparse Hamiltonians seem to be mathematically easier to handle than local ones. One direction follows immediately from
Theorem 1 by observing that local Hamiltonians are in particular sparse. The other direction, namely that
adiabatic computations with sparse Hamiltonians can be simulated efficiently by standard quantum computers, follows from the implementation of sparse Hamiltonians using quantum circuits presented in [21].
From the algorithmic point of view, these result show that all quantum algorithmic questions can be
framed in the language of eigenstates and spectral gaps of local or even sparse Hermitian matrices. The
design of quantum algorithms can thus draw on the wide scientific literature on these fundamental objects,
in particular expander theory [27], rapidly mixing Markov chains (see, e.g., [28, 29]), and mathematical
physics. This raises the hope that tools from these areas might be useful to tackle the difficult challenge
of designing new quantum algorithms. Indeed, probability theory was already used in analyzing spectral
gaps of Hamiltonians (see, e.g., [30] and references therein, as well as the proofs in this paper). In addition,
we note that the adiabatic model appears ideally suited for the task of quantum state generation, which was
shown recently to be essential for many important quantum algorithmic problems [21].
As for experimental applications, our result shows that universal quantum computation can in principle
be implemented adiabatically. We bring the model one step closer to physical realization by showing that
adiabatic computation with a more physically realistic set of Hamiltonians is also quantum universal:
Theorem 3. The model of adiabatic computation with two-body nearest neighbor Hamiltonians operating on six-state particles set on a two dimensional grid, is polynomially equivalent to standard quantum
computation.
Experimentally realizing quantum computation using the adiabatic model has potential advantages from
the point of view of fault tolerance, i.e., reliable computation in the presence of noise. [20] argues that if
adiabatic computers were cooled below the Hamiltonian’s energy gap, decoherence could essentially be
ruled out. Furthermore, [20] studies the effect of unitary control errors in the Hamiltonian and conclude
that such errors might, in fact, help the adiabatic computation. Our result combined with these studies
—such as simulated annealing— though the problem of [16, 17] can be solved efficiently by other classical methods.
2
indicates that the possibility of fault tolerant adiabatic computation deserves to be further studied, both
theoretically and experimentally. Experimental realization of small-scale adiabatic computation was already
demonstrated [31].
Finally, we consider further the possible relevance of the above results to fault tolerance. If the Hamiltonian of the adiabatic evolution leaves a certain subspace invariant, it actually suffices to lower bound the
spectral gap of the Hamiltonian restricted to that subspace. (This is the case in Subsection 3.1) However,
the full spectral gap, namely the gap in the entire Hilbert space, may be important in the context of noise,
especially thermal fluctuations (see [20]) 2 . It turns out that
Theorem 4. The above results hold with Hamiltonians with inverse polynomial “full” spectral gap.
1.2 Key Ideas
Given a quantum circuit [35], we associate with it a corresponding adiabatic computation. Without loss of
generality we will assume that the input to the quantum circuit consists of n qubits all initialized to |0i’s 3 .
Then, a sequence of L unitary gates, U1 , . . . , UL , each operating on one or two qubits, is applied to the state.
The system’s state after the ℓ’th gate is |α(ℓ)i. The output of the quantum circuit is in general a complicated
quantum state |α(L)i of n qubits, which is then measured in the standard basis.
A natural attempt to design an adiabatic computation that mimics the circuit’s computation would be
to define Hfinal to be a local Hamiltonian with |α(L)i as its ground state. This poses a difficulty: we want
to explicitly specify the Hamiltonian without knowing the complicated output of the quantum circuit in
advance.
The key to solving this problem is based on an idea by Kitaev [36] and Feynman [37]. Instead of
designing a Hamiltonian that has the final unknown state of the circuit as its ground state, a task that seems
impossible, one can define Hfinal to be a local Hamiltonian whose ground state is the entire history of the
quantum computation, in superposition:
L
|ηi :=
√
X
1
|α(ℓ)i ⊗ |1ℓ 0L−ℓ ic .
L + 1 ℓ=0
(1)
The right (L qubits) register is a clock which counts the steps by adding 1s from left to right. The superscript
c denotes clock qubits. For simplicity, denote |γℓ i := |α(ℓ)i ⊗ |1ℓ 0L−ℓ ic . Kitaev [36] defined a Hamiltonian
Hfinal involving five body interactions (three clock particles and two computation particles) that has |ηi as
its ground state. The idea is that the unary representation of the clock enables a local verification of correct
propagation of the computation from one step to the next. For the initial Hamiltonian Hinit we require that
it has |γ0 i as its unique ground state. H(t) is taken to be a convex combination of Hinit and Hfinal .
A technical problem lies in showing that the spectral gap of the intermediate Hamiltonian H(t) is larger
than 1/L2 . To do this, we use a mapping of the Hamiltonian to a Markov chain corresponding to a random
walk on the time steps. We then apply the conductance bound from the theory of rapidly mixing Markov
chains [28] to bound the spectral gap of this chain. We note that in general, applying the conductance bound
requires knowing the limiting distribution of the chain, which in our case is hard since it corresponds to
knowing the coefficients of the ground state at all times. We circumvent this problem by noticing that it is
actually sufficient in our case to know very little about the limiting distribution of the Markov chains, namely
that it is monotone (in a certain sense to be defined later.). This allows us to apply the conductance bound,
and deduce that the spectral gap is Ω(1/L2 ). From this is follows that if T ≫ L4 , the adiabatic system
2
The importance of non-negligible spectral gaps to fault tolerance appeared already in geometric and topological quantum
computation [32–34]. Note that these models differ from adiabatic computation in that the entire computational space has the
same energy, and hence the spectral gap is irrelevant from an algorithmic point of view
3
Otherwise, the first n gates can be used to flip the qubits to the desired input.
3
will end up close to the history state |ηi. Extracting the output of the quantum circuit from the history state
efficiently is easy: Measure all qubits and if the clock is at state |1l i, the computational qubits carry the
result of the circuit. Otherwise, start from scratch 4 .
This scheme would not suffice to prove Theorem 3. The basic problem lies in arranging sufficient interaction between the computational and the clock particles, since if the particles are set on a grid, each clock
particle can only interact with four neighbors. We circumvent this problem as follows. Instead of having
separate clock and computational particles, we now assign to each particle both clock and computational
degrees of freedom (this is what makes our particles six-states). We then construct a computation that propagates locally over the entire set of particles, snaking up and down each column of the lattice. The adiabatic
evolution would now end up in the history state of this snake-like sequence of states.
Organization of Paper: In Section 2 we describe the model and state some relevant facts about Markov
chains. Section 3 shows how adiabatic systems with local Hamiltonians allowing five- and later three-body
interactions, can simulate efficiently conventional quantum computations. Section 4 shows how to adapt the
construction to a two-dimensional grid. We conclude with open questions.
2 Preliminaries
2.1 The Adiabatic Computation Model
For background on n-qubit systems, quantum circuits and Hamiltonians, see [35]. Consider a quantum
d
|ψ(t)i = H(t)|ψ(t)i. H(t),
system composed of n particles, governed by Schrödinger’s equation: −ih̄ dt
called the Hamiltonian of the system,
is
a
Hermitian
matrix.
It’s
eigenvalues
are
called energies. We require
P
A
that H(t) is local, i.e., H(t) = A H (t) where A runs over constant size subsets of the particles, and
H A (t) operates trivially on all but A (i.e., it is a tensor product of a Hamiltonian on A with identity on the
particles outside of A). The system is initialized in a tensor product state, the ground state (lowest energy
eigenstate) of the initial Hamiltonian Hinit = H(0). One then slowly modifies the Hamiltonian over a time
of length T from Hinit to Hfinal = H(T ) by setting H(t) := (1 − t/T )Hinit + (t/T )Hfinal (from now
on we use s = t/T ). In the limit of large T we are guaranteed by the adiabatic theorem [22, 23] that the
final state will be very close to the final ground state. Just how large T should be is determined by the
spectral gap of the time dependent Hamiltonian H(s), denoted ∆(H(s)), which is the difference between
the Hamiltonian’s lowest and next to lowest eigenvalue. More precisely, the adiabatic theorem requires that
!
k dH
ds k
.
(2)
T = Ω
mins∈[0,1] {∆2 (H(s))}
In fact, if there exists a subspace S such that H(t) leaves S invariant for all t, and the initial groundstate
belongs to S, then it is sufficient that the above condition holds for H(t) restricted to S. For the adiabatic
algorithm to be efficient, we require that T is polynomial in n, i.e., bounded by nc for some constant
c. Typically, the norm of dH
ds is bounded by some small constant (2, in the cases we study). We therefore
require that for all s ∈ [0, 1] the spectral gaps ∆(H(s)) are at least 1/nc for some (preferably small) constant
c. If this condition holds, the final state is close to the ground state of Hfinal , which can be thought of as the
outcome of the computation. In the end, the particles are measured to give the result of the computation.
2.2 Markov Chains and Hermitian Matrices
Under certain conditions, there exists a standard mapping of Hermitian matrices to Markov chains (i.e.,
stochastic matrices). Let G be a Hermitian matrix operating on an L + 1 dimensional Hilbert space. Assume
4
This gives an overhead factor of L, which can be avoided by adding, say, 10L identity gates to the quantum circuit at the end,
so that most of the history state is concentrated on the final state |α(L)i)
4
that all the entries of G are non-negative. Consider the eigenvector (α0 , . . . , αL ) of G, with the largest
eigenvalue µ. Assume µ > 0 and αi > 0 for all 0 ≤ i ≤ L. We can now define the matrix P by:
Pij :=
αj
Gij .
µαi
(3)
P is well defined and is stochastic because all its entries are non-negative and each of its rows sums up to
one. It is easy to verify the following fact:
Fact 1. (v0 , . . . , vL ) is an eigenvector of G with eigenvalue δ if and only if (α0 v0 , . . . , αL vL ) is a left
eigenvector of P with eigenvalue δ/µ.
We will consider Hermitian matrices of the form G = cI −H for some constant c and some Hamiltonian
H. The above fact implies that if (α0 , . . . , αL ) is the groundstate of H with eigenvalue λ then (α20 , . . . , α2L )
is the limiting distribution of P (i.e., left eigenvector with maximal eigenvalue 1), and the gap between P ’s
largest and second largest eigenvalues is equal to ∆(H)/(c − λ). Another useful fact is the following; its
proof is based on the Perron-Frobenius theorem:
Fact 2. [38] Let G be Hermitian, with non-negative entries, and ∃k < ∞ s.t. all entries of Gk are strictly
positive. Then G’s largest eigenvalue is positive and non-degenerate. Moreover, all entries of the corresponding eigenvector are positive.
2.3 Spectral Gaps of Markov Chains
Numerous techniques were developed to bound the spectral gap of a Markov chain matrix, as it is related [29]
to the important quantity of the mixing time of the chain. In this paper we use the conductance bound [28].
Given a stochastic matrix P
P with limiting distribution π, and a subset B ⊆ {0, . . . , L}, P
the flow from
:=
:=
B is given by: F (B)
i∈B,j ∈B
/ πi Pij . For each B, define the π-weight as π(B)
i∈B πi . The
conductance of P is defined by
ϕ(P ) := min
B
F (B)
,
π(B)
where we minimize over all non-empty subsets B ⊆ {0, . . . , L} with π(B) ≤ 12 .
Theorem 5. (The conductance bound [28]): The eigenvalue gap of P is ≥ 21 ϕ(P )2 .
3 Equivalence of Adiabatic and Quantum Computation
We show here how to simulate a quantum circuit with L two-qubit gates on n qubits by adiabatic computation on n + L qubits. The initial ground state in the adiabatic evolution is |γ0 i = |0n i ⊗ |0i, and the final
state is, as explained in the introduction, the history state defined in Equation 1. We start by allowing five
qubit interactions, and later show how to reduce it to three.
3.1 Universality of Adiabatic Computation with Five Qubit Interactions
3.1.1
The Hamiltonian
We would like to define a Hamiltonian Hinit that has as its ground state |γ0 i, and Hfinal that has as its ground
state |ηi. To do this, we construct local Hamiltonians which “check” that the ground states are the correct
5
ones, and assign energy penalty whenever this is not the case. The following Hamiltonian checks that the
clock’s state is indeed ℓ = 0 in the beginning
Hclockinit = |1ih1|c1 ,
where the subscript indicates which clock qubits the projection operates on. We also define Hclock , which
checks that the clock’s state is always of the form of |1ℓ 0L−ℓ ic by assigning an energy penalty to any basis
state on the clock qubits that contains the sequence 01:
L−1
X
Hclock :=
ℓ=1
|01ih01|cℓ,ℓ+1 ,
The ground space S of Hclock is spanned by exactly those states that represent a legal clock state. Next
consider Hinput , which checks that the input of the computational qubits is all zeroes, by penalizing all
states with clock state ℓ = 0 whose computation qubits are not all zero:
Hinput :=
n
X
i=1
|1ih1|i ⊗ |0ih0|c1 .
We define:
Hinit := Hclockinit + Hinput + Hclock
(4)
Claim 6. |γ0 i is a ground state of Hinit with eigenvalue 0.
Proof. The fact that Hinit|γ0 i is easy to verify. All eigenvalues of Hinit are non negative since all terms in
Hinit are positive semi definite.
Hfinal is defined by checking that the propagation from step ℓ − 1 to ℓ is correct, i.e., corresponds to the
application of the gate Uℓ :
L
Hfinal :=
1X
Hℓ + Hinput + Hclock
2
(5)
ℓ=1
where for 1 < ℓ < L, the following Hamiltonian corresponds to the desired check:
Hℓ :=
I ⊗ |100ih100|cℓ−1,ℓ,ℓ+1 − Uℓ ⊗ |110ih100|cℓ−1,ℓ,ℓ+1
.
−Uℓ† ⊗ |100ih110|cℓ−1,ℓ,ℓ+1 + I ⊗ |110ih110|cℓ−1,ℓ,ℓ+1
(6)
The three-qubit terms in Hl can be seen as |lihl|, |l − 1ihl − 1|, |lihl − 1| and |l − 1ihl| respectively, i.e.,
corresponding to moving one step forward or backwards. For the boundary cases ℓ = 1, L, we omit one
clock qubit from these terms:
H1 := I ⊗ |10ih10|1,2 + I ⊗ |00ih00|1,2 − U1 ⊗ |10ih00|1,2 − U1† ⊗ |00ih10|1,2
(7)
HL := I ⊗ |11ih11|L−1,L + I ⊗ |10ih10|L−1,L − UL ⊗ |11ih10|L−1,L − UL† ⊗ |10ih11|L−1,L .
We remark that for the results in this subsection (and only here) the terms Hclock and Hinput can be completely omitted. We introduce them here for the sake of consistency with the rest of the paper.
Claim 7. |ηi is the ground state of Hfinal with eigenvalue 0.
Proof. Same as that of Claim 6.
6
3.1.2
The Spectral Gap
It remains to show that the spectral gap of H(s) is non-negligible. It is easy to verify that:
Claim 8. The subspace S0 , spanned by |γ0 i, . . . , |γL i is invariant under H(s), i.e., H(s)(S0 ) ⊆ S0
Hence, by Subsection 2.1, Theorem 1 follows from
Claim 9. ∆(HS0 (s)) = Ω(L−2 ) for all s ∈ [0, 1].
(Here and later we use the notation HA to denote the Hamiltonian restricted to a subspace A.)
Proof. Let us write the Hamiltonians HS0 ,init and HS0 ,final in the basis |γ0 i, . . . , |γL i of S0 . Hclock and
Hinput are 0 on S0 and can thus be ignored. We have the following (L + 1) × (L + 1) matrices:


0 0 ... 0
 0 1 ... 0 


(8)
HS0 ,init =  . . .
. ,
 .. .. . . .. 
0 0 ... 1
HS0 ,final =
1
1
1
1
2 |γ0 ihγ0 | − 2 |γ0 ihγ1 | − 2 |γL ihγL−1 | + 2 |γL ihγL |
L−1
X
+
(− 12 |γl ihγl−1 | + |γl ihγl | − 21 |γl ihγl+1 |)
l=1





= 




1
2
1
−2
0
−1
2
1
1
−2
0
−1
2
..
.
1
..
0
1
−2
..
.
.
.
.
0
0
···
.
1
−2
0
···
0
.
.
.
.
.
.
.
0
..
.
1
1
−2
0
.
..
.
.
.
.
.
−1
2
1
1
−2
0
−1
2
1
2





.




(9)
We now lower bound ∆(HS0 ). We consider two cases:
The case s < 1/3: Here, HS0 (s) is sufficiently close to HS0 ,init which has a spectral gap of 1, and we can
use standard techniques
(Gerschgorin’s Circle Theorem [39]) to show that the gap of HS0 (s) is larger than
P
1/3. Let Ri = j6=i |HS0 (s)ij |. The Circle Theorem states that for each i there is an eigenvalue in the
disk of radius Ri around HS0 (s)ii . For s < 1/3, HS0 (s)0,0 ≤ 1/6 and R0 < 1/6, so the corresponding
eigenvalue is less than 1/3. Similarly, we obtain that the eigenvalue corresponding to HS0 (s)L,L is at least
2/3. For all 0 < i < L, Ri < 1/3, and HS0 (s)ii = 1, and so the remaining eigenvalues are also at least
2/3. Hence the gap is larger than 1/3.
The case s ≥ 1/3: We note that HS0 ,final is the Laplacian of the simple random walk [29] of a particle on
a line of length L + 1. A standard result in Markov chain theory implies ∆(HS0 ,final = Ω(1/L2 ) [29]. For
large enough s, the matrix has enough weight of a random walk to apply Markov chain techniques.
Let (α0 , . . . , αL ) be the ground state of HS0 (s) with the eigenvalue λ. Since the spectral norm of HS0 (s)
is at most 2, λ is at most 2 in absolute value. Define the Hermitian matrix G(s) = 4I − HS0 (s). Clearly,
G(s) has non-negative entries and for large enough k, all entries of G(s)k are strictly positive. Hence,
following Fact 2, we obtain that the largest eigenvalue µ = 4 − λ of G(s) is positive and non-degenerate
and the corresponding eigenvector (α0 , . . . , αL ) has positive entries. We can now map the matrix G(s) to a
stochastic matrix P (s) as described in Subsection 2.2. The transition matrix P (s) describes a random walk
on the line of L + 1 sites:
7
Pk,k−1 Pk,k+1
0
1
...
k−1
k
k+1
...
L
Figure 1: The random walk corresponding to P (s)
Fact 3. For all s > 0, the ground state of HS0 (s) is monotone, namely α0 ≥ α1 ≥ . . . ≥ αL > 0.
Proof. It is easy to check that G(s) preserves positivity and monotonicity, and (α0 , . . . , αL ) is the limit
of the application of G(s)/µ on the monotone vector (1, . . . , 1) infinitely many times. (This is because all
other eigenvalues of G(s) are strictly smaller than µ in absolute value, so their contribution decays to 0.)
Hence, the limiting distribution of P (s), π = (α20 , . . . , α2L ), is monotone. We use this and simple
combinatorial arguments to prove the following claim, the proof of which can be found in Appendix A:
Claim 10. ϕ(P (s)) ≥
1
36(L+1)
By Theorem 5, we have that the spectral gap of P (s) is larger than 2·(36)2 1·(L+1)2 . We have by Subsection
2.2 that ∆(HS0 ) ≥ 2·(36)2µ(L+1)2 . Finally, notice that µ = 4 − λ ≥ 2 since λ ≤ 2.
Remark In fact, we can prove a lower bound on the full gap: ∆(H(s)) = Ω(L−3 ), using the terms Hclock
and Hinput . The proof follows the first part of the proof of Claim 11 and is omitted here.
3.2 From Five Qubits to Three
To move to three-body interactions, we modify Hl in Equations 6,7, by leaving in the expressions corresponding to clock qubits only those terms which correspond to the current time step ℓ:
Hℓ′ := I ⊗ |0ih0|cℓ − Uℓ ⊗ |1ih0|cℓ − Uℓ† ⊗ |0ih1|cℓ + I ⊗ |1ih1|cℓ .
(10)
This introduces terms in the Hamiltonian which interact between legal and non-legal clock states, and so
H(s) no longer leaves the subspace S0 invariant. To this end, we assign a much larger energy penalty to
states outside of the legal clock states. Set J = O(L9 ) and define:
L
′
Hinit
:= Hclockinit + J · Hclock + Hinput ,
′
Hfinal
1X ′
:=
Hℓ + J · Hclock + Hinput
2
(11)
ℓ=1
Clearly, the new Hamiltonians have the same ground states as the old ones. Theorem 1 follows from:
Claim 11. ∆(H ′ (s)) = Ω(L−3 ) for all s ∈ [0, 1].
Proof. (Sketch. See appendix.) The proof builds on the proof of Claim 9, with two additional ideas. First,
we restrict attention to states with legal clock states, i.e., states in S, and observe that the Hamiltonian HS′ (s)
can be block diagonalized, where one of the blocks is exactly HS0 (s) from Claim 9. We show that the lowest
eigenvalue in the other blocks is Ω(L−3 ) because of Hinput which is non zero there. This shows the desired
spectral gap for HS′ (s). Since S is not invariant, we need to consider states outside of S. We use similar
ideas to those used in [40] in the context of quantum NP-complete problems: since the energy given to states
outside of S by Hclock is at least J, they cannot appear in the ground state of the Hamiltonian with large
weights, and the interaction with them does not effect the lower energy states that much.
8
4 Two-Body Local Interactions on a Two-Dimensional Lattice
In this section we prove Theorem 3. First, assume without loss of generality that the quantum circuit consists
of R rounds, where each round is composed of n gates (some can be the identity gate), as in Figure 2. This
can be done by introducing extra identity gates.
|0i
|0i
|0i
|0i
Figure 2: The first gate in each round is a one-qubit gate applied to the first qubit. For i = 2, . . . , n, the i’th
gate is a two-qubit gate applied to qubits i − 1 and i.
The adiabatic computation is performed on 6-dimensional particles, arranged on a two-dimensional
square lattice with n rows and R + 1 columns. Each column will correspond to a round of the computation
of the circuit. The six internal states of each particle are divided to four groups, corresponding to four “clock”
N
degrees of freedom: “unborn”: |i, “first phase”:|i,
↑ |i,
↓ “second phase”: |i,
↑⋄ |i,
↓⋄ and “dead”:| i.
The two “first phase” states and two “second phase” states correspond to computational degrees of freedom,
namely to the “zero” and “one” state of a qubit. We write |i
↑↓ (and similarly |i)
↓↑⋄ to denote a general state
that belongs to the subspace spanned by |i
↑ and |i.
↓
We now define the states |γl i. See Figure 4 for illustration. Once again, |γ0 i corresponds to the all |0i
input to the quantum circuit. Here, it is the state in which the particles in the leftmost column are in |i
↑
whereas the rest are “unborn”: |i. |γℓ i transforms to |γℓ+1 i by changing the particles on the lattice in
the following order (which resembles a propagation of a snake): first, we modify the particles in the left
most column from top to bottom, by applying the first round of computation on their internal qubit degree of
freedom, while simultaneously changing the particle’s state from a first-phase state to a second-phase state.
Once we reach the last particle in the column, we start moving up the same column, copying the qubit’s
state of the particle one site to the right, leaving the copied particle in the “dead” phase, while the particle
to the right is now in its first phase. When we reach the topmost particle in the column, all particles in this
column are dead and the particles in the next column are ready for the next round of computation. Finally,
in |γL i all columns but the last one are “dead”, and the n qubits in the right column are in the superposition
corresponding to the final state of the quantum circuit, i.e., the superposition obtained by mapping |0i to ↑
and |1i to .
↓ We get L = 2(R + 1)n states (two for each site in the lattice), plus the initial state.
As in the non-geometrical adiabatic computation, we now use the |γℓ i to define the initial and final
groundstates: the initial groundstate
of the adiabatic computation is |γ0 i and the final one is the history state
PL
1
|γ(ℓ)i.
|ηi defined by |ηi = √L+1
ℓ=0
4.1 The Hamiltonian
The notion of a legal clock state now means a state whose clock degrees of freedom has the same shape as
one of the states in Figure 4. The following claim is easy verification (see also Table 1).
Claim 12. A state of n × R + 1 six state particles set in an n × (R + 1) grid is of one of the shapes as in
Figure 4 if none of the following forbidden configurations appear in the state:
N
N
N N
,
↓ ,
↑
⋄ , ↑
↓
↓ , ↑
↓↑⋄ ,
, ↑↓ ,
↑↓ ↑↓ ,
↓↑⋄ ↓↑⋄ ,
↑↓ ↓↑⋄ ↓↑⋄
N
N
N
↓
↑
↑↓
↓↑⋄
↑↓
,
,
,
,
, N , N ,
.
,
⋄
↑
↓
⋄
↑
↓
⋄
↑
↓
↑↓
9
We define a two body nearest neighbor Hamiltonian which forbids the aboveN
configurations. For example, if the rule forbids A = (i, j)N
in state N
to the left of B = (i, j + 1) in state , then the corresponding
term in the Hamiltonian is (|, ih, |)A,B . Summing over all the forbidden configurations in Claim
12 applied to all pairs of particles, We have
X
Hclock := J
Hr
r∈rules
where J =
O(L−9 ).
Hinput checks that none of particles in the leftmost column are in |i:
↓
Hinput :=
n
X
i=1
(|ih
↓ |)
↓ i,1 .
We can now define the initial Hamiltonian to be:
Hinit := (I − |ih
↑ |
↑ − |ih
↓ |)
↓ 1,1 + Hinput + Hclock .
The first term checks that the top left particle is in a |i
↑↓ state. For Hfinal we have:
L
Hfinal :=
1X
Hℓ + Hinput + Hclock .
2
ℓ=1
As before, the terms Hℓ check the propagation from |γℓ i to |γℓ+1 i. Since |γℓ i differs from |γℓ+1 i only in
two adjacent lattice sites, this is a two-body nearest neighbor Hamiltonian. There are two types of steps: an
(u)
(d)
“upward” step and a “downward” step, and thus Hℓ := Hℓ + Hℓ . For the upward step, in which all that
is supposed to happen is that the particle moves to the right and does not change its internal qubit state, the
Hamiltonian is:
N
N
N
N
(u)
:= | , ih
↑
, |
↑ + |,
⋄ ih,
↑
↑⋄ | − | , ih
↑ ,
↑⋄ | − |,
↑⋄ ih , |
↑ +
Hℓ
N
N
N
N
| , ih
↓
, |
↓ + |,
⋄ ih,
↓
↓⋄ | − | , ih
↓ ,
↓⋄ | − |,
↓⋄ ih , |.
↓
where the leftNparticle is followed by the right particle. The first
Nline corresponds to changing the state
|,
↑⋄ i into | , i.
↑ The second line is similar for |,
↓⋄ i and | , i.
↓
(d)
For the downward step, Hℓ needs to check that a two-qubit gate is applied correctly. We denote the
upper particle involved in this gate first, followed by the lower particle in the gate. Restricted to the four
(d)
|,
↓↑⋄ i
↑↓ states and to the four |,
⋄ i
↑
↓
⋄ states, Hℓ is the following 8 × 8 matrix:
↑
↓
I
−U
(d)
:=
Hℓ
−U † I
(d)
(cf. Equation 6); everywhere else Hℓ is zero.
The Hamiltonians Hℓ for the edge cases, i.e., the top most and bottom most particles of each column,
are defined similarly, where the only difference is that the first gate of a round in the quantum circuit is a
one-qubit gate, and so Hℓ operates only on one particle. Formally, if U is the corresponding gate, we define
Hl as the 4 × 4 matrix
I
−U
Hl :=
−U † I
operating on the two states |i
↓ and the two states |i.
↑
↓↑⋄ It is easy to check that
Claim 13. |γ0 i,|ηi are the ground states of Hinit, Hf inal , respectively, with eigenvalues 0.
It remains to prove that the spectral gap is non-negligible:
Claim 14. ∆(H(s)) ≥ 1/L3 for all s ∈ [0, 1]
Proof. The proof is essentially identical to that of Claim 11. Note that here too S is not invariant.
10
5 Conclusions and Open Questions
This paper demonstrates that quantum computation, its strengths and weaknesses, can be studied and implemented entirely within the adiabatic computation model, without losing its computational power. These
results raise many open questions in various directions. Can the connections to new pools of techniques, in
particular Markov chain Monte Carlo methods and adiabatic evolutions, be helpful in improving our understanding of the computational power of quantum systems, and in designing new algorithms? An important
intermediate question is to try to understand known quantum algorithms in the adiabatic language; as we
showed all known algorithms have efficient adiabatic representations.
A full fault tolerance theory for the adiabatic computation model is yet to be developed. In particular,
experimental study of the susceptibility of adiabatic computations to decoherence and errors might be of
great importance. Improving the parameters presented in this work, and in particular, making the simulation
of quantum systems by adiabatic evolutions linear instead of polynomial, as well as decreasing the dimensionality of the particles from six to two or three, might be very important for implementation applications.
6 Acknowledgments
We wish to thank Dave Bacon, Ed Farhi, Leonid Gurvitz and Umesh Vazirani for inspiring discussions.
DA’s effort is supported in part by ARO grant number DAAD19-03-1-0082, NSF ITR grant number CCR0121555. WvD work is supported in part by funds provided by the U.S. Department of Energy (DOE)
and cooperative research agreement DF-FC02-94ER40818, by a CMI postdoctoral fellowship, and by an
earlier HP/MSRI fellowship. JK’s effort is partly sponsored by the Defense Advanced Research Projects
Agency (DARPA) and Air Force Laboratory, Air Force Material Command, USAF, under agreement number
F30602-01-2-0524 and FDN00014-01-1-0826. OR’s effort is supported by the Army Research Office grant
DAAD19-03-1-0082 and NSF grant CCR-9987845. Part of this work was done while DA, WvD, JK, ZL
were members of the Mathematical Sciences Research Institute, Berkeley, CA, and OR was at the Institute
for Advanced Study, Princeton, NJ.
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A
Proof of Claim 10
For the proof, we consider two cases, depending on whether the set B contains 0 or not.
• If 0 ∈ B: Let k be the smallest index such that k ∈ B but k+1 ∈
/ B. Clearly, F (B) ≥ πk P (s)k,k+1 =
αk αk+1 G(s)k,k+1 /(4 − λ). We bound each of these terms. We have αk+1 , αk ≥ √ 1
, since by
2(L+1)
1
2
1
2
≤ (L + 1)α2k+1 .
+ ··· +
we have the bound ≤
π(B) ≤ and the monotonicity of
Also, using the definition of G and the assumption that s ≥ 1/3 we get that G(s)k,k+1 ≥ 16 , which
1
1
, and since 2 ≤ 4 − λ ≤ 6, F (B)/π(B) ≥ 36(L+1)
.
gives the lower bound F (B) ≥ 12(4−λ)(L+1)
α2i ,
13
α2k+1
α2n
• If 0 ∈
/ B, we let k be the smallest index such that k ∈
/ B but k + 1 ∈ B. Because of the
2
2
monotonicity of αi , we know that π(B) ≤ αk+1 + · · · + α2L ≤ α2k+1 (L + 1). Again, we have
F (B) ≥ αk αk+1 G(s)k,k+1 /(4 − λ) and hence F (B)/π(B) ≥ 61 αk αk+1 /((4 − λ)α2k+1 (L + 1)) ≥
1
1
6(4−λ)(L+1) ≥ 36(L+1) .
Together these show the desired bound.
B Proof of Spectral Gap For Three Qubit Interactions
Here we prove Claim 11. We first consider HS′ (s). From now on we omit the prime. This Hamiltonian is
block diagonal as in Figure 3.
H S0
0
H S1
HS2n−1
0
HF
Figure 3: The Hamiltonian H(s) can be block diagonalized as above.
In the first subspace is S0 we get exactly the same Hamiltonian which we have dealt with in Claim 9.
In addition, we have many subspaces, denoted Sj , which can be viewed as the analogues of S0 . S0 consists
of the set of states which the quantum circuit reaches starting from the all zero input. Sj consists of the set
of states which the quantum circuit reaches starting from the j’th input, i.e., the n bit binary representation
of j ∈ {0, . . . , 2n − 1}. The Sj ’s span S. We first show
Claim 15. The spectral gap of HS (s) is Ω(1/L3 ).
Proof. By Claim 9, it suffices to argue that the energy of HS (s) on each of the Sj for j 6= 0 is at least 1/L3 .
This is true because of the penalty assigned by Hinput , but the argument is slightly subtle since Hinput
applies only for the first clock’s state. We have HSj (s) = HS0 (s) + HSj ,input . HSj ,input is diagonal, with
its top left element at least 1 (it actually equals the number of 1’s in the binary representation of j) and all
other diagonal elements are zero. Hence, we can lower bound the ground energy of HSj (s) with the ground
energy of the Hamiltonian HS0 (s) + H ′ where


1 0 ... 0
 0 0 ... 0 


H ′ :=  . . .
. .
 .. .. . . .. 
0 0 ... 0
We can now use a geometrical lemma by Kitaev (Lemma 14.4 in [36]) that bounds the ground energy of
the sum of two Hamiltonians. It states that if the spectral gaps of both Hamiltonians are larger than Λ, and
the angle between the two ground spaces is θ, then the ground energy of the sum is at least 2Λ sin2 (θ/2). In
our case, the spectral gaps are Ω(L2 ) by Claim 9. Moreover, the angle between the ground states satisfies
cos(θ) ≤ 1 − 1/L by the monotonicity of the ground state of HS0 (s). It follows that the ground energy of
HSj (s) is Ω(1/L3 ) for all Sj with j 6= 0.
14
We now want to consider the entire Hilbert space, including states outside of S0 . We use similar ideas
to those used in [40] in the context of quantum NP-complete problems. The following lemma allows us to
eliminate the interaction between S and its orthogonal space which we denote F, and essentially consider
only the Hamiltonian restricted to the space S while ignoring the F space completely. Once restricted to the
space S, the spectral gap analysis follows from Claim 15.
We would like to say that since the energy given to states in F is so high, they cannot appear in the
ground state of the Hamiltonian with large weights, and the interaction with them does not effect the lower
energy states that much. This is captured by the following general lemma, where we assume that J, the
penalty for being in the forbidden subspace, is much larger than the size of the interaction between the
forbidden subspace and the valid subspace. K in the following upper bounds this interaction, and thus we
assume J ≫ K in the following lemma.
Lemma 16. Let H = H1 +H2 be the sum of two Hamiltonians operating on some Hilbert space H = S +F.
The Hamiltonian H2 is such that all its eigenvectors are inside either S or F. Moreover, the eigenvalues of
eigenvectors in S are 0 and those of eigenvectors in F are at least J. Also, assume that kH1 k ≤ K for some
K ≥ 1. Assume that the Hamiltonian HS,1 , i.e., the restriction of H to the space S, has an eigenvector
|ηi ∈ S with eigenvalue 0 and all other eigenvectors are with eigenvalues at least 1. Then, the Hamiltonian
′
3/4
1/4 and all other
H has an eigenvector |η ′ i with eigenvalue at most 0 such
√ that k|η i − |ηik ≤ 8K /J
3/2
eigenvectors are with eigenvalues at least 1 − 70K / J.
Proof. Any vector |vi ∈ H can be written as |vi = α1 |v1 i + α2 |v2 i with |v1 i ∈ S, |v2 i ∈ F, α1 , α2 ∈ R,
α1 ≥ 0 and α21 + α22 = 1.
Fact 1:
hv|H|vi + K
α22 ≤
J
2
Proof: hv|H|vi ≥ hv|H2 |vi − kH1 k ≥ Jα2 − K.
Fact 2:
hv|H|vi ≥ hv1 |H1 |v1 i − 4K|α2 |
Proof: Write
hv|H|vi ≥ hv|H1 |vi = (1 − α22 )hv1 |H1 |v1 i + 2α1 α2 Rehv1 |H1 |v2 i + α22 hv2 |H1 |v2 i
≥ hv1 |H1 |v1 i − K(2α22 + 2|α2 |)
using α1 ≤ 1. The result follows with |α2 | ≥ α22 .
In the first part of the proof we will show that |η ′ i exists and that it’s very close to |ηi. First,
hη|H|ηi = hη|H1 |ηi + hη|H2 |ηi = hη|H1 |ηi = 0
where we used |ηi ∈ S. Therefore, the Hamiltonian H must have an eigenvector |η ′ i whose eigenvalue is
at most 0. Write
|η ′ i = α1 |η1′ i + α2 |η2′ i
q
2
2
′
′
with |η1 i ∈ S, |η2 i ∈ F, α1 , α2 ∈ R such that α1 ≥ 0, α1 + α2 = 1 and use Fact 1 to obtain |α2 | ≤ K
J
and then Fact 2 together with |η1′ i ∈ S to get
K 3/2
4 √ ≥ hη1′ |H1 |η1′ i = hη1′ |ΠH1 Π|η1′ i.
J
Write
|η1′ i = γ1 |ηi + γ2 |η ⊥ i
15
and let us assume that we have chosen the phase of |η ′ i such that γ1 ≥ 0. Then hη1′ |ΠH1 Π|η1′ i =
γ22 hη ⊥ |ΠH1 Π|η ⊥ i ≥ γ22 using ΠH1 Π|ηi = 0 and the assumption on ΠH1 Π.
Finally, we get that
k|η ′ i − |ηik ≤ k|η ′ i − α1 |η1′ ik + kα1 |η1′ i − |η1′ ik + k|η1′ i − γ1 |ηik + kγ1 |ηi − |ηik ≤
p
√
K/J + K/J + 2K 3/4 /J 1/4 + 4K 3/2 / J ≤ 8K 3/4 /J 1/4
√
where we used α1 ≥ α21 ≥ 1 − K/J and γ1 ≥ γ12 ≥ 1 − 4K 3/2 / J.
In the second
part of the proof we will show that for any vector |ξi orthogonal to |η ′ i, hξ|H|ξi ≥
√
1 − 70K 3/2 / J . As before, we write
|ξi = β1 |ξ1 i + β2 |ξ2 i
with |ξ1 i ∈ S, |ξ2 i ∈ F, β1 , β2 ∈ R such that β12 + β22 = 1. Then from Fact 1,
β22 ≤
hξ|H|ξi + K
J
Therefore, if β22 ≥ 2K/J we are done. Assume that β22 < 2K/J. From Fact 2 and our assumptions on
ΠH1 Π it follows that
p
hξ|H|ξi ≥ hξ1 |H1 |ξ1 i − 4 2K 3 /J
p
= hξ1 |ΠH1 Π|ξ1 i − 4 2K 3 /J
p
≥ 1 − |hη|ξ1 i|2 − 4 2K 3 /J
Now, using hη ′ |ξi = 0 and |ηi ∈ S:
|hη|ξ1 i| = |hη|ξi| = |hη|ξi| − |hη ′ |ξi| ≤ |hη − η ′ |ξi| ≤ k|ηi − |η ′ ik ≤ 8
K 3/4
J 1/4
Finally, by combining the inequalities above, we obtain
√
√
√
√
hξ|H|ξi ≥ 1 − 64K 3/2 / J − 4 2K 3/2 / J ≥ 1 − 70K 3/2 / J
Corollary 17. Let H = H1 + H2 be the sum of two Hamiltonians operating on some Hilbert space H =
S + F. The Hamiltonian H2 is such that all its eigenvectors are inside either S or F. Moreover, the
eigenvalues of eigenvectors in S are 0 and those of eigenvectors in F are at least J. Also, assume that
kH1 k ≤ K for some K ≥ 1. Assume that the Hamiltonian HS,1 has an eigenvector |ηi ∈ S with eigenvalue
a and all other eigenvectors are with eigenvalues at least b. Then, the Hamiltonian H has an eigenvector
|η ′ i with eigenvalue at most a such that
k|η ′ i − |ηik ≤ 8
(K + a)3/4
√
J 1/4 b − a
√
and all other eigenvectors are with eigenvalues at least b − 70(K + a)3/2 / J.
Proof. Define H1′ = (H1 − a · I)/(b − a), H2′ = H2 /(b − a) and H ′ = H1′ + H2′ . Notice that kH1′ k ≤
(K +a)/(b−a) and that H2′ eigenvalues on F are at least J/(b−a). Then, using the above lemma, we obtain
that H ′ has an eigenvector with eigenvalue at most 0 whose distance from |ηi is at most 8((K + a)/(b −
a))3/4 /(J/(b − p
a))1/4 . Moreover, all other eigenvectors of H ′ are with eigenvalues at least 1 − 70((K +
a)/(b − a))3/2 / J/(b − a). We now complete the proof by translating these results back to H.
16
We apply Corollary 17 by taking H2 to be Hclock and H1 to be the remaining terms such that H(s) =
H1 + H2 . Notice that kH1 k ≤ O(L) because it consists of O(L) terms, each of constant norm. Also note
that the ground energy a of HS,1 = HS (s) is at most 21 . Moreover, we have b − a ≥ 1/L3 by the same proof
as in the spectral gap of the non-geometrical Hamiltonian. Hence, Corollary 17 implies that the spectral gap
decreases by at most
√
√
70(K + a)3/2 / J = O(L3/2 / J).
Moreover, the distance between the ground state of H(1) and |ηi is at most
8
(K + a)3/4
√
= O(L3/4 /J 1/4 ).
1/4
J
b−a
By choosing J = O(L9 ) we obtain that the spectral gap of H(s) is at least 1/(2L3 ) and that the ground
state of H(1) is close to |ηi. Hence, adiabatic simulation in time O(L6 ) is possible.
17
C
The States |γℓ i For the Two Dimensional Construction
a)
c)
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
↓
↑
↓
↑
↓
↑
↓
↑
↓
↑
↓
↑
↓
↑
↓
N ↑
↓
↑
N ↓
↑
⋄
↑
↓
⋄
↑
↓
⋄
↑
↓
⋄
↑
↓
N
b)
d)
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
↓↑⋄
↓↑⋄
↓↑⋄
↑↓
↑↓
↑↓
↑↓
N
N
N
N
N
N
N
↑↓
↑↓
↑↓
↑↓
↑↓
↑↓
↑↓
Figure 4: a) The state at the beginning of the r’th round. The n particles in the rth column are in their “first
phase”, where their computational degrees of freedom are in the state of the circuit’s qubits in the beginning
of the rth round. b) Next, the gates of the rth round of the circuit are applied on the qubits from top to
bottom. The particles on which the gates have been applied are now in their “second phase.” c) Next, the
particle’s computational state are moved one column to the right, and their phase becomes “first” again. The
internal qubit-state of the particles does not change in this phase. d) Once this phase is over, we are back to
a state of the same shape we had in a), one column (and one round) further.
D
Rules for the Clock Hamiltonian on the Lattice
18
Forbidden configuration
N
,
↓ ,
↑
⋄ ↑
↓
N N N
,
↓ ,
↑
⋄
↑
↓
N N
, ↓ ,
↑
↓ ↑
↓ ,
↑
⋄ ↑
↓
⋄ ,
↑
↓
↓ ↑
⋄
↑
↓
⋄
↑
↓
N
↓
↑
,
,
⋄
↑
↓
⋄
↑
↓
⋄
↑
↓
↓
↑
↓
↑
↓
↑
,
, N
⋄
↑
↓
N
N
,
N
⋄
↑
↓
,
↓
↑
Guarantees that
is to the right of all other qubits
N
is to the left of all other qubits
N
and
are not horizontally adjacent
only one of ,
↑↓ ↓↑⋄ per row
only ↓↑⋄ above ↓↑⋄
only ↑↓ below ↑↓
and
N
are not vertically adjacent
no below ↓↑⋄ and no ↑↓ below
N
Table 1: Local rules that guarantee that a basis state is in S
19